SOLUTION: determine whether quadrilateral MATH is a parallelogram, a rectangle, a rhombus, or a square given the vertices M (5,4), A (3, -6), T ( 0, -10), and H (2,0). Note: I keep comin

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Question 554928: determine whether quadrilateral MATH is a parallelogram, a rectangle, a rhombus, or a square given the vertices M (5,4), A (3, -6), T ( 0, -10), and H (2,0). Note: I keep coming up with an answer that doesn't make sense, the other problems, no problem.
Found 2 solutions by Edwin McCravy, KMST:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Plot the points and draw the figure MATH:

 M(5,4), A(3,-6), T(0,-10), H(2,0). 



Lines are parallel if they have the same slope.  A parallelogram is
a quadrilateral with both pairs of oposite sides parallel.

So we use the slope formula on all four sides.

m = 

Slope of MA:   M (5,4), A (3,-6), 

m =  =  = 5

Slope of TH:   T(0,-10), and H(2,0). 

m =  =  = 5

So one pair of sides are parallel since both have slope 5.

Slope of MH:    M(5,4), H(2,0).

m =  =  = 

Slope of AT:    A(3,-6), T(0,-10),

m =  =  =  = 
 
So the other pair of sides are also parallel since both have slope .

So MATH is a parallelogram.

Edwin
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
We can calculate slopes of MA, AT, TH, HM, MT, and AH
For MA, slope =
For AT, slope =
For TH, slope =
For HM, slope =
The fact that opposite sides of quadrilateral MATH have the same slope, means those pairs of opposite sides are parallel. That proves that it is a parallelogram.
If the adjacent sides were perpendicular, we would have four right angles, and it would be a rectangle (or maybe even that special kind of rectangle that we call square). If the sides were perpendicular, the product of their slopes would be -1.
However, is not -1. So, there are no right angles in MATH. Math is not a square or a rectangle.
Could it be a rhombus? If it were a rhombus, the diagonals would be perpendicular.
Let's calculate the slope of the diagonals
For MT, slope =
For AH, slope =
The product of the slopes, , is not -1, so the diagonals are not perpendicular, and MATH is not a rhombus.
Quadrilateral MATH is a parallelogram. It is neither a square, nor a rectangle, nor a rhombus.
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